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Integral Calculus> ∫cos(2 cot-1{ root(1-x/1+x)} = ????...

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∫cos(2 cot-1{ root(1-x/1+x)} = ????

Nikhitha H Jagadeesh , 11 Years ago

Grade 12

2 Answers

Jitender Singh

Ans:
Hello Student,
Please find answer to your question below

$I = \int cos(2cot^{-1}\sqrt{\frac{1-x}{1+x}})dx$
$x = cos\theta$
$dx = -sin\theta d\theta$
$I = \int cos(2cot^{-1}\sqrt{\frac{1-cos\theta }{1+cos\theta }})(-sin\theta )d\theta$
$I = \int cos(2cot^{-1}(tan\frac{\theta }{2}))(-sin\theta )d\theta$
$I = \int cos(2cot^{-1}(cot(\frac{\pi }{2}-\frac{\theta }{2}))(-sin\theta )d\theta$
$I = \int cos(\pi -\theta )(-sin\theta )d\theta$
$I = \int -cos(\theta )(-sin\theta )d\theta$
$I = \frac{1}{2}\int 2cos(\theta )(sin\theta )d\theta$
$I = \frac{1}{2}\int (sin2\theta )d\theta$
$I = \frac{-cos2\theta }{4} + c$
$I = \frac{-cos(2cos^{-1}x) }{4} + c$

Last Activity: 11 Years ago

Ritika Das

Hello I am a student.

I = -cos(2cos^-1x)/4 + c should have been further expressed as follows:-

I = -(2cos²theta-1)/4 + c

or, I = -(2x²-1)/4 + c

or, I = -(x²)/2 + c

Last Activity: 8 Years ago

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